I'd like to think Starfleet didn't make that guy resign just because of his half-romulan deal
The problem is Simon Tarses knew he was half-Romulan, rather than half-Vulcan. He lied on his application to Starfleet. That's why he would have been punished.
While he certainly would have gotten some kind of reprimand, I doubt they booted him out. For one thing, Starfleet has let people get away with a lot worse, up to and including attempted (and even successful!) murder. Lying on an application about something that is completely irrelevant to his suitability for the job is a small blemish on what looks like an otherwise good career record. We even see another character with exactly the same crime who gets off scot free - Julian Bashir.
For another, it would look absolutely horrible. Picard would make as stink and under the circumstances his desires would have quite a bit of weight. I imagine that if Tarses fought the discharge his lawyer and the press would have a field day. "My client hid his heritage because he feared illegal discrimination based on race, and then the moment a witch-hunting admiral and her mind-raping minion discovered the truth poor Crewman Tarses was immediately thrown out because of that very reason!" Maybe that explains Starfleet's willingness to sweep the Bashir case under the rug as well.